Re: Anti-diagonalist page

On 4 Sep 2005 02:18:54 -0700, "William of Ockham"
<d3uckner@xxxxxxxxxxxxxx> wrote:

>Ullrich---
>The author of the text I was refuting was not denying the
>existence of an uncountable set.
>---
>
>I asumed he was. Note he also puts scare quotes around 'sets'.
>
>---
>Anyway, the point is that Slater is not objecting to
>the truth of the fact which mathematicians express
>by saying that the cardinality of the reals is greater
>than the cardinality of the naturals.
>---
>
>On the assumption that "the cardinality of the reals is greater than
>the cardinality of the naturals" requires that infinite sets exist, I
>assume he would be objecting to this.

Assume what you want. I didn't assume anything, I read what he wrote:

"Cantor offered several proofs
that there is no one-one correlation between the real numbers and the
natural numbers, but ***only the presumption that there are infinite
numbers can turn whatever impossibility there is here into a seeming
demonstration that the number of the real numbers is greater than the
number of natural numbers***."

He doesn't object to the validity of the proof that there is no
"one-one correlation". And _by definition_ that is synonymous
with saying that the cardinality of the reals is greater than
the cardinality of the natural numbers.

>I'm basing this on more of
>Slater's writing than the brief quotation I gave on this list. For
>example, his paper 'Grammar and Sets' which is to appear in the
>*Australasian Journal of Philosophy* next March. Apologies if this was
>given out of context.
>
>---
>He's objecting to _saying_ that this entails that there are "infinite
>numbers" of various sorts. Which is silly, because _that_ is just a
>matter of definition.
>---
>
>Even if I have misunderstood his point, there is still the objection
>that you can't define something ('number') that already has a meaning.

Yes, that seemed to be his point. I _pointed out_ that that seemed
to be his point, in that post that you seem to have refuted
before reading! Here:

>> It's like he thinks that
>> the meaning of a technical term in mathematics is a priori
>> and forever exactly the same as its meaning in common
>> speech. Simply stupid.

Yes, there still is that objection. And that objection is
completely and utterly stupid - many terms in mathematics
means something other than what they mean in ordinary
speech. You say there's still this objection, like it
proved he was right or I hadn't answered his main point
or something. No, this objection is precisely one of
several reasons why that excerpt you posted is simply
dumb.

>One may agree with "the cardinality of the reals is greater than the
>cardinality of the naturals" but disagree that "cardinality" means
>"number" in the ordinary sense. That is also W.'s point.

Huh?????????? Who has every suggested that "cardinality"
meant the same as "number" in the ordinary sense?

>---
>To give another, perhaps better reply to this: If we insist
>on making an analogy with unicorns the analogy would be
>this: Slater says that yes, the proof of the existence
>of horses with horns _is_ valid, but he objects to inferring
>from that that unicorns exist. Which is simply silly.
>Once we have accepted that horses with horns do exist
>_then_ the existence of unicorns _is_ just a matter of
>definition.
>---
>
>Agree, and apologies if I misunderstood your point here. I was dealing
>with three long replies at once.
>
>---
>Lemme try to summarize for you. Slater says yes, the proof
>that there is no bijection from the naturals to the reals
>is valid, but he insists that this is not evidence for the
>existence of "infinite numbers" of various sizes. To make
>this coherent we need to write "cardinal" for "infinite number".
>If we do that then yes, the theorem does prove that there
>are infinite numbers of various sizes, by definition of
>"cardinality". Insisting otherwise is just silly.
>---
>
>No, it proves that there are cardinals of various sizes.

You really haven't read my original post _yet_, right?
Even more amazing, you didn't even read the paragraph
above.


************************

David C. Ullrich
.

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